Mild solution of a mixed problem for a third-order semilinear hyperbolic equation with the wave operator

Authors

  • Viktor I. Korzyuk Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus , Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surganava Street, Minsk 220072, Belarus
  • Jan V. Rudzko Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surganava Street, Minsk 220072, Belarus

Keywords:

non-linear hyperbolic equation of the third-order, mixed problem, generalised solution, mild solution
Supporting Agencies
The study was carried out with the financial support of the Moscow Center for Fundamental and Applied Mathematics (Agreement No. 075-15-2025-345).

Abstract

For a semilinear hyperbolic equation of the third-order given in the first quadrant we study a mild solution of a mixed problem in which the initial conditions are specified on the spatial half-line and the mixed conditions are specified on the time half-line. The operator in the equation is a composition of the wave operator and the transport operator. Mild solution is defined as a solution to coupled integral equations that are satisfied by a classical solution. It is shown that under some smoothness conditions on the initial and boundary data the problem under consideration admits the existence and uniqueness of the mild solution. It is established that the twice continuously differentiable mild solution is the limit of classical solutions to the problem under study.

Author Biographies

  • Viktor I. Korzyuk, Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus, Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surganava Street, Minsk 220072, Belarus

    doctor of science (physics and mathematics), academician of the National Academy of Sciences of Belarus, full professor; professor at the department of bio- and nanomechanics, faculty of mechanics and mathematics, Belarusian State University, and chief researcher at the department of differential equations, Institute of Mathematics, National Academy of Sciences of Belarus

  • Jan V. Rudzko, Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surganava Street, Minsk 220072, Belarus

    junior researcher at the department of differential equations

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Published

2026-01-04

Issue

No. 3 (2025): Journal of the Belarusian State University. Mathematics and Informatics

Section

Differential Equations and Optimal Control

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How to Cite

[1]
Korzyuk, V.I. and Rudzko, J.V. 2026. Mild solution of a mixed problem for a third-order semilinear hyperbolic equation with the wave operator. Journal of the Belarusian State University. Mathematics and Informatics. 3 (Jan. 2026), 15–28.